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  "body": "\n## Summary\n\nThe unitary dual $\\widehat{G}$ of a [[locally compact group|AHV05ZBCYW]] $G$ is the set of equivalence classes of irreducible [[unitary representations|74BPYJNZTM]] of $G$.\nIt usually carries the Fell topology.\nIt is the space on which a \"Fourier transform\" of functions on $G$ lives.\nFor abelian groups it is the Pontryagin dual, a group again.\nFor nonabelian groups it is only a topological space, often non-Hausdorff.\nDetermining it for a given group is a central problem of representation theory.\n\n## Explanations\n\n### L1 human\n\nFor the circle, the irreducible unitary representations are the characters $z \\mapsto z^n$, one for each integer $n$, so the unitary dual of the circle is $\\Z$.\nThis is why functions on the circle have Fourier series indexed by integers.\nFor the real line the characters are $x \\mapsto e^{ixy}$, one for each real $y$, which gives the Fourier transform.\nFor a general group the unitary dual plays the same role: it lists the basic frequencies in which functions on the group can be decomposed.\n",
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